Singular Chain of a Hyperplane.

Question

I refer to the definitions of Hatcher's Algebraic Topology. Is it possible to model a hyperplane $H$ (or half of it) of $\mathbb{R}^n$ with a singular chain? And if - how would its boundary look like? How do singularities effect the boundary?

Answer 1

As stated in the comments, by compactness this is not possible. But there is a setting where it makes sense to take certain infinite formal sums of chains; these represent elements of Borel-Moore homology rather than ordinary singular homology.